Complex dynamical behavior of a three-order memristive Murali–Lakshmanan–Chua circuit

In this paper, we investigate complex dynamical behaviors of a three-order modified MLC circuit, such as the coexistence of grazing bifurcations without parameters, the coexistence of infinite attractors, and transient chaos. Due to a piecewise function, the circuit system is a typical non-smooth system. Firstly, after we give the analytic solution of system equation in every smooth region, we derive analytic conditions for the existence of stable periodic solutions and grazing bifurcations in these regions. At the same time, we also find initial conditions when periodic orbits and grazing bifurcations exist. The results show that we have conducted research on the circuit system in therms of theory. Secondly, by choosing appropriate parameters, we verify the correctness of theory through numerical simulations. Furthermore, when initial conditions are changed, we obtain the coexistence of grazing bifurcations. As it relies on changes in initial conditions, it is called grazing bifurcation without parameters, which is rarely reported in prior research on other systems by now. When choosing same parameters but different initial conditions, we present different attractors, which means the coexistence of multiple attractors, including the coexistence of different chaotic attractors and infinite periodic orbits, as well as the coexistence of quasi-periodic orbit and infinite periodic orbits. It is noted that infinite coexisting attractors are rarer than finite coexisting attractors, and these infinite attractors in this paper have relation with periodic orbits in smooth regions. To our surprise, when parameters and initial conditions are all the same, the trajectory converges to chaos for a long time, then stabilizes to a periodic solution, which means the appearance of transient chaos. In a word, grazing bifurcation without parameters causes abrupt change of the state, the coexistence of attractors leads to initial value-dependent multistability, and transient chaos prolongs the transient process. These three factors increase system complexity and uncertainty together. Studying these phenomena not only has theoretical significance but also holds important practical value for improving the performance, reliability, and functionality of modern circuits.